We prove that the natural isomorphism$\mathbb {F}_{2^{h}}\cong \mathbb {F} _{2}^{h}$induces a bijection between stabilizer codes on n quqits with local dimension$q=2^{h}$and binary stabilizer codes on hn qubits. This allows us to describe these codes geometrically: a stabilizer code over a field of even order corresponds to a so-called quantum set of symplectic polar spaces. Moreover, equivalent stabilizer codes have a similar geometry, which can be used to prove the uniqueness of a$[\![{4,0,3}]\!]_{4}$stabilizer code and the nonexistence of both a$[\![{7,1,4}]\!]_{4}$and an$[\![{8,0,5}]\!]_{4}$stabilizer code.